Recent progress on deep learning (DL)-based medical image segmentation can enable fast extraction of clinical parameters for efficient clinical workflows. However, current DL methods can still fail and require manual visual inspection of outputs, which is time-consuming and diminishes the advantages of automation. For clinical applications, it is essential to develop DL approaches that can not only perform accurate segmentation, but also predict the segmentation quality and flag poor-quality results to avoid errors in diagnosis. To achieve robust performance, DL-based methods often require large datasets, which are not always readily available. It would be highly desirable to be able to train DL models using only small datasets, but this requires a quality prediction method to ensure reliability. We present a novel segmentation framework utilizing an ensemble of deep convolutional neural networks with Monte Carlo sampling. The proposed framework merges the advantages of both state-of-the-art deep ensembles and Bayesian approaches, to provide robust segmentation with inherent quality control. We successfully developed and tested this framework using just a small MRI dataset of 45 subjects. The framework obtained high mean Dice similarity coefficients (DSC) for segmentation of the endocardium (0.922) and the epicardium (0.942); importantly, segmentation DSC can be accurately predicted with low mean absolute errors (≤0.035), in the absence of the manual ground truth. Furthermore, binary classification of segmentation quality achieved a near-perfect accuracy of 99%. The proposed framework can enable fast and reliable medical image analysis with accurate quality control, and training of DL-based methods using even small datasets.